New quadratic polynomials with high densities of prime values
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Abstract
Hardy and Littlewood’s Conjecture F implies that the asymptotic density of prime values of the polynomials <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f Subscript upper A Baseline left-parenthesis x right-parenthesis equals x squared plus x plus upper A comma"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>f</mml:mi> <mml:mi>A</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:msup> <mml:mi>x</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>+</mml:mo> <mml:mi>x</mml:mi> <mml:mo>+</mml:mo> <mml:mi>A</mml:mi> <mml:mo>,</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">f_A(x) = x^2 + x + A,</mml:annotation> </mml:semantics> </mml:math> </inline-formula> <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A element-of double-struck upper Z"> <mml:semantics> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo> ∈ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">Z</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">A \in \mathbb {Z}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , is related to the discriminant <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Delta equals 1 minus 4 upper A"> <mml:semantics> <mml:mrow> <mml:mi mathvariant="normal"> Δ </mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mo> − </mml:mo> <mml:mn>4</mml:mn> <mml:mi>A</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\Delta = 1 - 4A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f Subscript upper A Baseline left-parenthesis x right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>f</mml:mi> <mml:mi>A</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">f_A(x)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> via a quantity <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C left-parenthesis normal upper Delta right-parenthesis period"> <mml:semantics> <mml:mrow> <mml:mi>C</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="normal"> Δ </mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>.</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">C(\Delta ).</mml:annotation> </mml:semantics> </mml:math> </inline-formula> The larger <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C left-parenthesis normal upper Delta right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>C</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="normal"> Δ </mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">C(\Delta )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is, the higher the asymptotic density of prime values for any quadratic polynomial of discriminant <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Delta"> <mml:semantics> <mml:mi mathvariant="normal"> Δ </mml:mi> <mml:annotation encoding="application/x-tex">\Delta</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . A technique of Bach allows one to estimate <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C left-parenthesis normal upper Delta right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>C</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="normal"> Δ </mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">C(\Delta )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> accurately for any <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Delta greater-than 0"> <mml:semantics> <mml:mrow> <mml:mi mathvariant="normal"> Δ </mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\Delta > 0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , given the class number of the imaginary quadratic order with discriminant <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Delta"> <mml:semantics> <mml:mi mathvariant="normal"> Δ </mml:mi> <mml:annotation encoding="application/x-tex">\Delta</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , and for any <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Delta greater-than 0"> <mml:semantics> <mml:mrow> <mml:mi mathvariant="normal"> Δ </mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:m
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it